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Properties

Label 2-75-15.8-c1-0-0
Degree $2$
Conductor $75$
Sign $0.229 - 0.973i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 1.22i)2-s + (−1.22 + 1.22i)3-s + 0.999i·4-s − 2.99·6-s + (1.22 − 1.22i)8-s − 2.99i·9-s + (−1.22 − 1.22i)12-s + 5·16-s + (−4.89 − 4.89i)17-s + (3.67 − 3.67i)18-s + 4i·19-s + (−2.44 + 2.44i)23-s + 3.00i·24-s + (3.67 + 3.67i)27-s − 8·31-s + (3.67 + 3.67i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.866i)2-s + (−0.707 + 0.707i)3-s + 0.499i·4-s − 1.22·6-s + (0.433 − 0.433i)8-s − 0.999i·9-s + (−0.353 − 0.353i)12-s + 1.25·16-s + (−1.18 − 1.18i)17-s + (0.866 − 0.866i)18-s + 0.917i·19-s + (−0.510 + 0.510i)23-s + 0.612i·24-s + (0.707 + 0.707i)27-s − 1.43·31-s + (0.649 + 0.649i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1/2),\ 0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904589 + 0.715908i\)
\(L(\frac12)\) \(\approx\) \(0.904589 + 0.715908i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
good2 \( 1 + (-1.22 - 1.22i)T + 2iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (4.89 + 4.89i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (2.44 - 2.44i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-7.34 - 7.34i)T + 47iT^{2} \)
53 \( 1 + (-9.79 + 9.79i)T - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 16iT - 79T^{2} \)
83 \( 1 + (2.44 - 2.44i)T - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.88591915642086371833069085244, −13.96125831356930491890603751287, −12.79465046856314394413945040041, −11.58250579457822642766888248788, −10.41457947495543246540209379939, −9.229114639916129374277969711062, −7.34620001755066011427968295329, −6.14944178014170683784485800338, −5.11986073900510071403803114577, −3.93615513183635762614798187508, 2.17607389815220050882504286441, 4.21091389250569490413296196158, 5.59619503202360186135513635054, 7.03391025821551306767908559308, 8.509438765201328517434046221591, 10.53061872040027379540653170350, 11.23859886643624843861479732933, 12.27147786681798035208451286844, 13.04551560405313689181081413683, 13.78849691301066602785506767810

Graph of the $Z$-function along the critical line